8.1 Exponential Function- base b is a positive number Asymptote- a line that a graph approaches as you move away from the origin Exponential Growth Factor-y=ab^x Growth Factor- the quantity 1 + r
8.2 Exponential Dacey Factor-f(x)=ab^x where a > 0 Decay Factor- The quantity 1 - r
8.3 Natural Base e (Euler number)- acts like e pie or i
8.4 Logarithum of y with base b-logby = x Common logarithm- The logarithum with base 10 Natural logarithum- The logarithum with base e
8.5 Different properties-Product, Quotient, & Power
8.6 Logarithmic equations-If the bases are equal then their exponents must be equal
8.7 Writing an exponential function with two points- plug one order pair in to find a then other order pair to solve for y =]
8.8 Logistic Growth Functions- a, c, and r are alll positive functions y= c
1 + ae ^-rx
8.1 practice
Identify the asymptote and y-intercept of the graph of the function
y=5x
x is to the first power causing the y-intercept to be at (0,1)
the bin the graph equals 0 so the asymptote is at the x-axis
8.2 practice
Tell whether the funtion is growth or decay
y=5(.25)^x
this is an example of decay because the base is less than 1
8.3 practice
simplify
2e^3 x e^4
2e^7
8.4 practice
rewrite in exponential form
log 100=x
10^x=100
x=2
8.5 practice
condense the function
ln(16)-ln(4)
ln(16/4)
ln(4)
8.6 practice
solve
5^x=8
log5 (8)=x
log(8)/log(5)=x
x=1.29
8.7 practice
write a exponential function of the form y=ab^x which passes through these points
(2,2) and (3,18)
2=ab^2
2/b^2=a
18=(2/b^2)b^3
18=2b
b=9
2/9^2=a
2/81=a
y=(2/81)9^x
8.8 practice
evaluate the funtion with 12/(1+5e^(-2x))
x=0
12/(1+5e^(-2 x 0))=y
12/(1+5e^0)=y
12/(1+5 x 1)=y
12/6=y
y=2
Career Applications:
USING A LOGARITHMIC SCALE WITH SOUND INTENSITY
Decibels (dB) are used to measure of sound intensity. They are often used with stereos and other sound emitting devices and these decibels are based on a log10 scale. The faintest noise that humans can hear is called the threshold of hearing. Its decibel number is really small, about a 0.3 billionths change in air pressure. The scale is given as dB = log (Number of times greater than threshold of hearing) × 10. A normal conversation is 60dB or, remembering to divide by the 10 from right of the formula, 106 = 1,000,000. Humans have a wide range of hearing for the conversation is a million times louder than the faintest sound we can hear and this is all measures with the use of logarithms.
CRYPTOGRAPHY AND GROUP THEORY
Cryptography is coding information in a certain way so that somebody could not intercept and decode a message written in the code. Today's methods use a simple mathematical device that is nearly impossible to decode, logarithms. Such codes cannot be unraveled by just looking at the figures in the messege, the formula for the code has to be generated, and that's difficult. For the maker of the code and the ones with a key, this messege is indeed readable, however, if anyone else tried to decode this encryption by intercepting the messeges between the two parties, he would have to invert the formula used to be able to read the information in the messeges. A set of functions that show these properties can be found in abstract mathematical research that studies relations between objects. This study is called group theory. Some groups have codes that act like logarithms or exponentials. Decoding the exponential codes is fairly easy, however, decoding the ones that act like logarithms is extremely hard and is under continued study to this day.
8.1
Exponential Function- base b is a positive number
Asymptote- a line that a graph approaches as you move away from the origin
Exponential Growth Factor- y=ab^x
Growth Factor- the quantity 1 + r
8.2
Exponential Dacey Factor- f(x)=ab^x where a > 0
Decay Factor- The quantity 1 - r
8.3
Natural Base e (Euler number)- acts like e pie or i
8.4
Logarithum of y with base b- logby = x
Common logarithm- The logarithum with base 10
Natural logarithum- The logarithum with base e
8.5
Different properties- Product, Quotient, & Power
8.6
Logarithmic equations- If the bases are equal then their exponents must be equal
8.7
Writing an exponential function with two points- plug one order pair in to find a then other order pair to solve for y =]
8.8
Logistic Growth Functions- a, c, and r are alll positive functions
y= c
1 + ae ^-rx
8.1 practice
Identify the asymptote and y-intercept of the graph of the functiony=5x
x is to the first power causing the y-intercept to be at (0,1)
the bin the graph equals 0 so the asymptote is at the x-axis
8.2 practice
Tell whether the funtion is growth or decay
y=5(.25)^x
this is an example of decay because the base is less than 1
8.3 practice
simplify
2e^3 x e^4
2e^7
8.4 practice
rewrite in exponential form
log 100=x
10^x=100
x=2
8.5 practice
condense the function
ln(16)-ln(4)
ln(16/4)
ln(4)
8.6 practice
solve
5^x=8
log5 (8)=x
log(8)/log(5)=x
x=1.29
8.7 practice
write a exponential function of the form y=ab^x which passes through these points
(2,2) and (3,18)
2=ab^2
2/b^2=a
18=(2/b^2)b^3
18=2b
b=9
2/9^2=a
2/81=a
y=(2/81)9^x
8.8 practice
evaluate the funtion with 12/(1+5e^(-2x))
x=0
12/(1+5e^(-2 x 0))=y
12/(1+5e^0)=y
12/(1+5 x 1)=y
12/6=y
y=2
Career Applications:
USING A LOGARITHMIC SCALE WITH SOUND INTENSITY
Decibels (dB) are used to measure of sound intensity. They are often used with stereos and other sound emitting devices and these decibels are based on a log10 scale. The faintest noise that humans can hear is called the threshold of hearing. Its decibel number is really small, about a 0.3 billionths change in air pressure. The scale is given as dB = log (Number of times greater than threshold of hearing) × 10. A normal conversation is 60dB or, remembering to divide by the 10 from right of the formula, 106 = 1,000,000. Humans have a wide range of hearing for the conversation is a million times louder than the faintest sound we can hear and this is all measures with the use of logarithms.CRYPTOGRAPHY AND GROUP THEORY
Cryptography is coding information in a certain way so that somebody could not intercept and decode a message written in the code. Today's methods use a simple mathematical device that is nearly impossible to decode, logarithms. Such codes cannot be unraveled by just looking at the figures in the messege, the formula for the code has to be generated, and that's difficult. For the maker of the code and the ones with a key, this messege is indeed readable, however, if anyone else tried to decode this encryption by intercepting the messeges between the two parties, he would have to invert the formula used to be able to read the information in the messeges. A set of functions that show these properties can be found in abstract mathematical research that studies relations between objects. This study is called group theory. Some groups have codes that act like logarithms or exponentials. Decoding the exponential codes is fairly easy, however, decoding the ones that act like logarithms is extremely hard and is under continued study to this day.